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Journal ArticleDOI

On a new functional transform in analysis: The maximum transform

21 Jun 1961-Bulletin of the American Mathematical Society (American Mathematical Society)-Vol. 67, Iss: 5, pp 501-503
TL;DR: In this paper, a convolution of two functions ƒ and g, h = ǫ * g, defined by g, is introduced, where h is a function defined by
Abstract: (1) F(xu x2, • • • , xN) = fi(xi) + f2(x2) + +/N(XN) over the region R defined by xi+x2 + • • • +XN = X> x^O. Under various assumptions concerning the ƒ», this problem can be studied analytically; cf. Karush [ l ; 2] , and it can also be treated analytically by means of the theory of dynamic programming [3]. I t is natural in this connection to introduce a \"convolution\" of two functions ƒ and g, h = ƒ * g, defined by

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Citations
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Journal ArticleDOI
TL;DR: In this paper, an algebraic approach to idempotent functional analysis is presented, which is an abstract version of the traditional functional analysis developed by V. P. Maslov and his collaborators.
Abstract: This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.

222 citations

Book ChapterDOI
01 Jan 2000
TL;DR: This Chapter also contains a discussion on operational semantics of the generally too abstract notion of membership function and a survey of variants of fuzzy sets and related matters.
Abstract: This paper is an introduction to fuzzy set theory. It has several purposes. First, it tries to explain the emergence of fuzzy sets from an historical perspective. Looking back to the history of sciences, it seems that fuzzy sets were bound to appear at some point in the 20th century. Indeed, Zadeh’s works have cristalized and popularized a concern that has appeared in the first half of the century, mainly in philosophical circles. Another purpose of the paper is to scan the basic definitions in the field, that are required for a proper reading of the rest of the volume, as well as the other volumes of the Handbooks of Fuzzy Sets Series. This Chapter also contains a discussion on operational semantics of the generally too abstract notion of membership function. Lastly, a survey of variants of fuzzy sets and related matters is provided.

201 citations


Cites background from "On a new functional transform in an..."

  • ...Operations on fuzzy intervals lead to consider max-based "convolutions" of pairs of functions, which have been previously considered by Bellman and Karush (1961), in the framework of dynamic programming....

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Journal ArticleDOI
TL;DR: In this paper, the authors employ the method of multipliers by Hestenes to resolve the dual gaps of engineering system design problems; it then develops an algorithmic procedure which preserves the separability characteristics of the system.
Abstract: For nonconvex problems, the saddle point equivalence of the Lagrangian approach need not hold. The nonexistence of a saddle point causes the generation of a dual gap at the solution point, and the Lagrangian approach then fails to give the solution to the original problem. Unfortunately, dual gaps are a fairly common phenomenon for engineering system design problems. Methods which are available to resolve the dual gaps destroy the separability of separable systems. The present work employs the method of multipliers by Hestenes to resolve the dual gaps of engineering system design problems; it then develops an algorithmic procedure which preserves the separability characteristics of the system. The theoretical foundations of the proposed algorithm are developed, and examples are provided to clarify the approach taken.

141 citations


Cites background from "On a new functional transform in an..."

  • ...Rearranging the terms in ( 5 ), the Lagrangian can be written as follows:...

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Journal ArticleDOI
TL;DR: In this paper, Fenchel and Moreau proved duality theorems for real linear topological spaces and their application to mathematical programming and the calculus of variations, as well as saddle point and optimal control problems.
Abstract: Let be a real linear topological space and its conjugate. We denote by the value of the linear functional on the element . For real functions on we introduce two operations: the ordinary sum and the convolution and also the transformation associating with its dual function on which is obtained from by the formula The following propositions hold.1) The operation is involutory: if and only if is a convex function and lower semicontinuous on .2) .3) Under certain additional assumptions These theorems were proved for a finite-dimensional space by Fenchel [93] and in the general case by Moreau [60].Chapter I is concerned with proving these theorems and generalizations of them.Chapter II is concerned with their application to mathematical programming and the calculus of variations. Proofs are given of very general duality theorems of mathematical programming and saddle point theorems. Constructions are then given which lead to extensions of optimal control problems, and an existence theorem is proved for these problems.Chapter III contains an investigation of problems of approximating and the set by an approximating set using methods of the theory of duality of convex functions. Duality theorems for some geometric characteristics of sets in are derived at the end of the chapter.

87 citations

References
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Book
21 Oct 1957
TL;DR: The more the authors study the information processing aspects of the mind, the more perplexed and impressed they become, and it will be a very long time before they understand these processes sufficiently to reproduce them.
Abstract: From the Publisher: An introduction to the mathematical theory of multistage decision processes, this text takes a functional equation approach to the discovery of optimum policies. Written by a leading developer of such policies, it presents a series of methods, uniqueness and existence theorems, and examples for solving the relevant equations. The text examines existence and uniqueness theorems, the optimal inventory equation, bottleneck problems in multistage production processes, a new formalism in the calculus of variation, strategies behind multistage games, and Markovian decision processes. Each chapter concludes with a problem set that Eric V. Denardo of Yale University, in his informative new introduction, calls a rich lode of applications and research topics. 1957 edition. 37 figures.

14,187 citations

Journal ArticleDOI
TL;DR: The relation between lost sales and inventory level is an important problem in inventory control as discussed by the authors, and an explicit mathematical solution is obtained by methods of general interest for a probabilistic model that arose in connection with consulting work for an industrial client.
Abstract: The relation between lost sales and inventory level is an important problem in inventory control. An explicit mathematical solution is obtained by methods of general interest for a probabilistic model that arose in connection with consulting work for an industrial client. Customer demand for a given commodity is a Poisson process with mean rate λ, and replenishment time for restocking is random. At any moment, the constant inventory n is divided between in-stock amount n0, and inreplenishment process amount n − n0. Customer arrival when n0 > 0 results in a unit sale and the initiation of replenishment of that unit. Successive replenishment times are independent. Customer arrival when n0 = 0, results in a lost sale. The unique stationary probabilities p(n0∣n) of the states n0 (fixed n), are obtained, they are given by the Erlang congestion formula, and depend upon the replenishment time only to the extent of its mean value. A generalization is obtained where λ may be a function of the state of the system. ...

70 citations


"On a new functional transform in an..." refers background in this paper

  • ...4) be defined by the equation (1) F(z) = max [e~f(x)]y z ^ O ....

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  • ...Let (1) f(x) = max [fi(x1)f2(x2) • • • /N(XN)], R...

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  • ...In the study of mathematical economics and operations research, we encounter the problem of determining the maximum of the function (1) F(xu x2, • • • , xN) = fi(xi) + f2(x2) + - - - +/N(XN) over the region R defined by xi+x2 + • • • +XN = X> x^O....

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  • ...Differentiating this relation with respect to xy we have dz dz dz (1) F\z) — = (ƒ'(*) - zf(x))e~ - xf(x)e~ — = - xf(x)e~ — • ax dx dx Hence, (2) x = - F(z)/F(z), or F'(z) + xF(z) = 0....

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  • ...Furthermore, since F(z) is unchanged when ƒ is replaced by its monotone envelope, we shall consider (1) only for monotone nondecreasing ƒ....

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Journal ArticleDOI
TL;DR: This paper presents an algorithm for obtaining the optimal allocation as a function of w, when the f i are arbitrary piecewise-linear, continuous functions, and is recursive and appropriate for hand or machine computation.
Abstract: The problem of distribution of effort is that of the optimal allocation of a given resource among various activities, that is, maximize ∑i=1n fi(xi) subject to ∑i=1n xi = w, xi ≥ 0. This paper presents an algorithm for obtaining the optimal allocation as a function of w, when the fi are arbitrary piecewise-linear, continuous functions. The method is recursive and appropriate for hand or machine computation. It is an extension of the well-known technique of ordering the segments of all the fi according to decreasing slope which applies in the special case when each fi has decreasing slopes (that is, is marginally decreasing).

29 citations